The New York Times reported on Saturday that Trump has privately questioned - to a senator and, more recently, an adviser - the authenticity of the 2005 recording on which he bragged about groping and kissing women without their consent.
As Mr. Moore has rejected calls to drop out even as more women have accused him of preying on them when they were teenagers, Republicans have given up any hope that he will fold his campaign.
At the beginning of the "Access Hollywood" video, Trump can be heard via a hot mic in a van.
Trump reportedly sees the calls for Moore to exit the Alabama race as a version of the scathing response that the "Access Hollywood" tape prompted.
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Sources say the president was also not happy with his daughter Ivanka Trump's quick reaction to the allegations against Moore. You know, I'm automatically attracted to beautiful - I just start kissing them. Grab 'em by the p***y. At the time, Trump acknowledged the voice on the recording was his and apologized for his vulgar comments about non-consensually grabbing women by the genitals while simultaneously dismissing his inappropriate language as "locker room talk".
"I said it, I was wrong, and I apologize", he said at the time.
Trump subsequently stated during a general election debate in October 2016 that he had never kissed or groped women without consent.
"But I couldn't get there", Trump said on the tape.
"I have great respect for women". The news operation's reputation took another hit more recently when it passed on Ronan Farrow's reporting about women alleging they had been sexually harassed and abused by Hollywood mogul Harvey Weinstein.